Step detection

In statistics and signal processing, step detection (also known as step smoothing, step filtering, shift detection, jump detection or edge detection) is the process of finding abrupt changes (steps, jumps, shifts) in the mean level of a time series or signal. It is usually considered as a special case of the statistical method known as change detection or change point detection. Often, the step is small and the time series is corrupted by some kind of noise, and this makes the problem challenging because the step may be hidden by the noise. Therefore, statistical and/or signal processing algorithms are often required. H [ m ] = ∑ i = 1 N ∑ j = 1 N Λ ( x i − m j , m i − m j , x i − x j , i − j ) {displaystyle H=sum _{i=1}^{N}sum _{j=1}^{N}Lambda (x_{i}-m_{j},m_{i}-m_{j},x_{i}-x_{j},i-j)}     (1) In statistics and signal processing, step detection (also known as step smoothing, step filtering, shift detection, jump detection or edge detection) is the process of finding abrupt changes (steps, jumps, shifts) in the mean level of a time series or signal. It is usually considered as a special case of the statistical method known as change detection or change point detection. Often, the step is small and the time series is corrupted by some kind of noise, and this makes the problem challenging because the step may be hidden by the noise. Therefore, statistical and/or signal processing algorithms are often required. The step detection problem occurs in multiple scientific and engineering contexts, for example in statistical process control (the control chart being the most directly related method), in exploration geophysics (where the problem is to segment a well-log recording into stratigraphic zones), in genetics (the problem of separating microarray data into similar copy-number regimes), and in biophysics (detecting state transitions in a molecular machine as recorded in time-position traces). For 2D signals, the related problem of edge detection has been studied intensively for image processing. When the step detection must be performed as and when the data arrives, then online algorithms are usually used, and it becomes a special case of sequential analysis.Such algorithms include the classical CUSUM method applied to changes in mean. By contrast, offline algorithms are applied to the data potentially long after it has been received. Most offline algorithms for step detection in digital data can be categorised as top-down, bottom-up, sliding window, or global methods. These algorithms start with the assumption that there are no steps and introduce possible candidate steps one at a time, testing each candidate to find the one that minimizes some criteria (such as the least-squares fit of the estimated, underlying piecewise constant signal). An example is the stepwise jump placement algorithm, first studied in geophysical problems, that has found recent uses in modern biophysics.